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Rita and Sam play the following game with n sticks on a [#permalink]
14 Jul 2008, 04:51

Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4, or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?

Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4, or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?

A. 7 B. 20 C. 22 D. 12 E. 16

I dont even understand the question!

Game theory on the GMAT? Interesting. Where is the question from?

The question defines a game- whoever removes the last stick wins. You can remove up to 5 sticks, so if you get to a situation where there are 5 or fewer sticks left and it's your turn, you are certain to win. If you are in a situation where there are 6 sticks and it's your turn, you have to remove at least one stick, and you can't remove all 6, so your opponent certainly is left with between 1 and 5 sticks, and certainly wins. So really what you'd want to do is force your opponent into a situation where he or she has 6 sticks. You can do that if you have between 7 and 11 sticks. And you will automatically have between 7 and 11 sticks on your turn if you force your opponent to have 12 sticks- which you can do if you have between 13 and 17 sticks, and so on. As long as you can force, after your turn, the number of sticks on the table to be a multiple of 6, you can be certain to win if you play correctly. The only way Rita loses, if she goes first, is if the number of sticks on the table is a multiple of six when the game starts. That is, for the answer choices, only D) defines a situation where Rita is certain to lose (as long as Sam plays well).

Or, the longer explanation- let's look at some of the possibilities here:

If n=7, and Rita goes first, Rita will remove just 1 stick. Sam is facing 6 sticks now, so whatever he does, Sam is going to lose, Rita is going to win.

If n=12, and Rita goes first, after Rita's turn, there will be between 7 and 11 sticks left. Sam can remove enough to leave Rita with 6 sticks, so Rita is going to lose, Sam is going to win.

If n=22, and Rita goes first, Rita will remove 4 sticks. There are 18 left. No matter what Sam does, he leaves between 13 and 17 on the table. Rita can now remove enough to leave 12 sticks. After Sam moves, there are between 7 and 11- Rita removes enough to leave 6 left. etc. Sam loses, Rita wins.

And so on.

_________________

Nov 2011: After years of development, I am now making my advanced Quant books and high-level problem sets available for sale. Contact me at ianstewartgmat at gmail.com for details.

i dont get it..do they HAVE to pick up a stick on each turn?..suppose if I see 6 sticks..i am not picking any..cause i know the opponent will win..so i just keep "passing" i.e not pick up a stick..

1. he who leaves the opponent with 6 sticks wins, as one cannot clear the table nor leave his opponent with >5 sticks. Same goes for any multiple of 6. 2. Rita starts, so it is as if John left Rita with as many sticks as there are on the table at start. 3. If those many sticks are 6 or a multiple of six, John wins.

I answer D. I think everyone else has done a great job of explaining what the question is asking for.

Jamesk486 wrote:

Rita and Sam play the following game with n sticks on a table. Each must remove 1, 2, 3, 4, or 5 sticks at a time on alternate turns, and no stick that is removed is put back on the table. The one who removes the last stick (or sticks) from the table wins. If Rita goes first, which of the following is a value of n such that Sam can always win no matter how Rita plays?

A. 7 B. 20 C. 22 D. 12 Rita goes first, if she takes 5, then 7 left, Sam takes 1, leaving 6 left, regardless of mow many Rita takes next, there is at most 5 left so Sam wins. If Rita takes only 1, then 11 left and Sam takes 5, leaving 6, Rita again, no matter what Rita does, Sam has at most 5 left and he can take them all. E. 16

I dont even understand the question!

_________________

------------------------------------ J Allen Morris **I'm pretty sure I'm right, but then again, I'm just a guy with his head up his a$$.

1. he who leaves the opponent with 6 sticks wins, as one cannot clear the table nor leave his opponent with >5 sticks. Same goes for any multiple of 6. 2. Rita starts, so it is as if John left Rita with as many sticks as there are on the table at start. 3. If those many sticks are 6 or a multiple of six, John wins.